By Chaohua Jia, Kohji Matsumoto

Comprises a number of survey articles on major numbers, divisor difficulties, and Diophantine equations, in addition to study papers on a number of facets of analytic quantity concept difficulties.

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**Example text**

Experiment. Math. 4 (l995), 169-173. [22] K. Kawada, Note on the sum of cubes of primes and an almost prime. Arch. Math. 69 (1997), 13-19. [23] K. Prachar , ~ b e ein r Problem vom Waring-Goldbach 'schen Typ 11. Monatsh. Math. 57 (1953) 113-116. 91 [24] K. F. Roth, Proof that almost all positive integers are sums of a square, a cube and a fourth power. J . London Math. Soc. 24 (1949), 4-13. [25] K. F. Roth, A problem in additive number theory. Proc. London Math. Soc. (2) 53 (1951), 381-395. [26] W.

Namely, s is either 1 or 2, and the natural numbers k and k j ( 0 5 j 5 s) are less than 6 . 3); C S i ( 9 ,a ) ns;,( 9 ,a ) e ( - a n / q ) , A(q,n) = ~ ( q ) - ~ - ' 8 ( p ,k ) 8 ( p ,k ) + 2, + 1, p e ( ~ 7 k ) > 2 7 and h 2 2, or when p 5 5 and Proof. 1, since k j < 6 (0 5 j 5 s). 3. Assume that ( p ,n). One also has T h e n one has B d ( p ,n ) = B(p,d) Prwf. 1) The series defining B d ( p ,n) and B ( p , n ) are finite sums in practice, because of the following lemma. 1. Let B(p, k ) be the number such that power of p dividing k , and let (iii) A d ( p ,n ) = A ( p , n ) = 0 , when p h 5.

Namely, s is either 1 or 2, and the natural numbers k and k j ( 0 5 j 5 s) are less than 6 . 3); C S i ( 9 ,a ) ns;,( 9 ,a ) e ( - a n / q ) , A(q,n) = ~ ( q ) - ~ - ' 8 ( p ,k ) 8 ( p ,k ) + 2, + 1, p e ( ~ 7 k ) > 2 7 and h 2 2, or when p 5 5 and Proof. 1, since k j < 6 (0 5 j 5 s). 3. Assume that ( p ,n). One also has T h e n one has B d ( p ,n ) = B(p,d) Prwf. 1) The series defining B d ( p ,n) and B ( p , n ) are finite sums in practice, because of the following lemma. 1. Let B(p, k ) be the number such that power of p dividing k , and let (iii) A d ( p ,n ) = A ( p , n ) = 0 , when p h 5.